So, what do you think? Was algebra invented? Was calculus discovered?
Invented for both.
But the characteristics of slope existed before they were described by humans…
So, you already have your answer?
Mathematics is a human construct. It relies on a set of axioms for validity. Who proposed these axioms before humans existed?
Even if we reify geometric axioms, they are nothing more than a subset of the possibly consistent mathematical axioms that can exist. While I hesitate to even reify the entire universe given the uncertain nature of quantum mechanics, to the extent that mathematical systems exist independent of man, every other system also exists equally, even the ones that are internally consistent but make absolutely no sense.
We just picked our current ones because they are the best at describing our universe.
I was just trying to provoke some thought on the subject is all. 
I looked up reify, but I’m still not sure what you’re talking about. Are you sauing the possibility of a thing presupposes its existence?
It doesn’t make any difference to the practicing mathematician, and there’s absolutely no evidence that would change anyone’s mind on the question.
I used it in the context of “elevating to the status of an object”. I.e. I am unsure of the ontological status of axioms, if they exist or not (as opposed to, say, matter, but I’m a bit iffy on those as well.)
Ah, the old constructionist/fallibilist vs. absolutist/Platonist debate.
My friend is a high school algebra teacher and he asked this question of his class. Here was the best response he got:
Deep. Very deep.
Emphasis mine.
That is just freakin’ scary. And not in a good way.
Discovered. My position is that if we ever run into intelligent alien life, they will agree that 1+1=2.
Granted, arithmetic isn’t exactly higher math, but the same basic philosophy applies.
Actually, didn’t two different people “invent” calculus independently? That seems to me to be evidence that it was discovered.
Arithmetic is a language; saying it was discovered is very much like saying English was discovered. After all, English, just like math, describes the real world, does it not?
We invented math, and have been discovering things about our invention ever since.
Works for me.
To an extent, but then again we have loads of mathematical languages yet only one single set of concepts. In my alien species hypothetical, would you not expect them to be versed in the same mathematical principles as us humans? Consider 12+8=20. The human languages that express this include:
12+8=20
XII+VIII=XX
C+8=15
01100+01000=10100
The way I see it, stuff like addition is a universal truth the we discovered, much like all of mathematics. Our math “language” is base 10, whereas aliens might use a system as different as roman numerals. The notation changes, but the concepts are the same.
Just like different languages have different words for “tree”, the tree wasn’t invented, it was discovered. The mathematical concepts weren’t invented, they were discovered.
The dressing we apply to the concepts is invented, but I wouldn’t even go too far with that. Suppose the aliens use base 3 for their computers. Would you then say that they invented base 3? I would not.
Tangentally related, are the laws of physics invented or discovered? I would say discovered. Math is used to describe these laws. The descriptions themselves – as in the marks on the paper – are invented, sure. But the mathematical concepts that describe the laws of physics are no more invented than the laws themselves.
We certainly didn’t invent gravity, nor did we invent the inverse square relationship of gravity’s effects. Right? Therefore, the math concept of “inverse square” had to have been discovered, not invented.
Maybe I’m getting ahead of myself. What, specifically, is the difference between an invention and a discovery?